With the rapid development of information processing technologies and communication technologies, documents have been digitized rapidly regardless of whether the documents are public or private. With the digitization of such documents, many individuals and companies have a considerable interest in security management of electronic documents. Countermeasures against tampering acts such as wiretapping or forgery of electronic documents have been actively studied in various fields in response to an increase in this interest. Regarding the wiretapping of electronic documents, security is ensured, for example, by encrypting the electronic documents. Further, regarding the forgery of electronic documents, security is ensured, for example, by using digital signatures. However, when the encryption or the digital signature to be used does not have high tampering resistance, sufficient security is not ensured.
The digital signature is used for specifying the author of an electronic document. Accordingly, the digital signature should be able to be generated only by the author of the electronic document. If a malicious third party is able to generate the same digital signature, the third party can impersonate the author of the electronic document. That is, an electronic document is forged by the malicious third party. Various opinions have been expressed regarding the security of the digital signature to prevent such forgery. As digital signature schemes that are currently widely used, a RSA signature scheme and a DSA signature scheme are known, for example.
The RSA signature scheme takes “difficulty of prime factorisation of a large composite number (hereinafter, prime factorisation problem)” as a basis for security. Also, the DSA signature scheme takes “difficulty of solving discrete logarithm problem” as a basis for security. These bases are based on that algorithms that efficiently solve the prime factorisation problem and the discrete logarithm problem by using a classical computer do not exist. That is, the difficulties mentioned above suggest the computational difficulty of a classical computer. However, it is said that solutions to the prime factorisation problem and the discrete logarithm problem can be efficiently calculated when a quantum computer is used.
Similarly to the RSA signature scheme and the DSA signature scheme, many of the digital signature schemes and public-key authentication schemes that are currently used also take difficulty of the prime factorisation problem or the discrete logarithm problem as a basis for security. Thus, if the quantum computer is put to practical use, security of such digital signature schemes and public-key authentication schemes will not be ensured. Accordingly, realizing new digital signature schemes and public-key authentication schemes is desired that take as a basis for security a problem different from problems such as the prime factorisation problem and the discrete logarithm problem that can be easily solved by the quantum computer. As a problem which is not easily solved by the quantum computer, there is a problem related to a multivariate polynomial, for example.
For example, as digital signature schemes that take the multivariate polynomial problem as a basis for security, those based on MI (Matsumoto-Imai cryptography), HFE (Hidden Field Equation cryptography), OV (Oil-Vinegar signature scheme), and TTM (Tamed Transformation Method cryptography) are known. For example, a digital signature scheme based on the HFE is disclosed in the following Non-Patent Literatures 1 and 2.